Elements of Polymer Structure and Viscoelasticity. David M. Parks Mechanics and Materials II February 18, 2004
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1 Elements of Polymer Structure and Viscoelasticity David M. Parks Mechanics and Materials II February 18, 2004
2 Outline Elements of polymer structure Linear vs. branched; Vinyl polymers and substitutions Packing of polymer chains Random/amorphous Glass transition temperature, T G Semi-crystalline Crystalline volume fraction; melting temperature Amorphous T G Elements of linear viscoelasticity Creep and relaxation Analogue models
3 Idealized Linear Elastic Response Linear elasticity: [NO] Creep [NO] Relaxation Note: units of E are stress.
4 Idealized Linear Viscous Response Linear viscosity: Creep: Relaxation: Note: units of η are stress time e.g., sec MPa The strain rate (and therefore stress) becomes arbitrarily large during an infinitesimal time interval, and then, like the strain-rate, goes to zero
5 Maxwell Model: an Idealized Linear Viscoelastic Response Creep: For times near the finite stress jump, all strain occurs in the elastic element; During the hold period, all strain occurs in the viscous element
6 Maxwell Model: an Idealized Linear Viscoelastic Response Relaxation: For times > 0, d ε(t)/dt=0 During the hold period, elastic strain is traded for viscous strain, and stress drops: Characteristic relaxation time: τ = η / E
7 Real Polymer Relaxation (an Idealization) Testing of real polymers under relaxation can be used to extract a time-dependent relaxation modulus, E r (t); Short-term response: E rg ; Long-term response: E re NOTE: provided ε 0 is sufficiently small (typically, less than 0.01), the relaxation modulus, E r (t), is approximately independent of ε 0.
8 Real Polymer Creep: (an Idealization) Testing of real polymers under suddenlyapplied constant stress can be used to extract a time-dependent creep function, J c (t); Short-term response: J cg ; Long-term response: J ce Note: units of J c (t) : 1/ stress NOTE: provided ε(t) remains sufficiently small (typically, less than 0.01), the creep function, J c (t), is approximately independent of σ 0.
9 Relaxation Modulus, E r (t) and Creep Function, J c (t): Inverse Functions of Time? QUESTION: Are these inverse functions? Is J c (t) E r (t) 1 for all times? ANSWERS: In general, they are not precise inverses. However, Equilibrium and glassy values are nearly inverse: For intermediate times, t. the error in assuming that they are inverse is typically only a few per cent at most
10 Linearity of Response (an Idealization) NOTE: similar linear scaling of stress relaxation response applies.
11 Superposition of Loading Suppose that the stress history input consists of a sequence of stress jumps, σ i, applied at successive times t i, with t 0 =0: THEN, the resulting strain history is given by
12 Special Case: Load/Unload
13 Correspondence Principle Suppose that a given load, P, produces displacement vector u(x) in a linear elastic body having Young s modulus E. The displacement vector depends on the position vector x = x e x + y e y + z e z. The magnitudes of the displacement and strain components are proportional to P and inversely proportional to E. EXAMPLE: Three-point Mid-span bending:
14 Correspondence Principle Now suppose that a given load jump, P(t), is applied to a geometrically identical linear viscoelastic body having creep function J c (t). All stress components in the body are time-independent, and spatially vary precisely as they do in an identical linear elastic body subject to the same load. The loading produces time-dependent displacement vector u(x,t) and corresponding strain components. The magnitudes of the displacement and strain components are proportional to both P and J c (t). EXAMPLE: Three-point Mid-span bending: For suddenly-applied load, replace 1/E with J c (t) in an elastic solution.
15 Correspondence Principle Suppose that a given displacement jump, (t), is applied to a geometrically identical linear viscoelastic body having stress relaxation modulus E r (t). All displacement and strain components everywhere in the body are time-independent, and precisely equal those in an identical linear elastic body subject to the same applied displacement. These boundary conditions produce time-dependent loads, P(t), and stresses The magnitudes of the time-dependent load and of the stress components are proportional to both and to E r (t). EXAMPLE: Three-point Mid-span bending: For suddenly-applied displacement, replace E with E r (t) in an elastic solution.
For an imposed stress history consisting of a rapidly applied step-function jump in
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